Planar U-frame (two columns + top beam), pinned at both ends — three load types on the top beam
This calculator finds the end reactions and end rotation of a planar three-bar U-frame using the classical elastic frame flexibility method (also called the force method or compatibility method). The frame consists of a left column of length l_1, a right column of length l_2, and a top horizontal beam of length l_3, with both column bases pinned (pin supports at A and B). The columns and beam each have their own flexural rigidity E_1I_1, E_2I_2, E_3I_3.
The method releases the horizontal reaction H_A and the end moment M_A at A as redundants, then writes two compatibility equations — one for the horizontal displacement and one for the rotation at A — in terms of the flexibility coefficients A_{HH}, A_{HM}, A_{MM} (geometry-only, independent of the load) and the load terms LP_H, LP_M (load-dependent). For a pinned-pinned frame (M_A = 0), only H_A is redundant, giving the closed-form result H_A = LP_H / A_{HH}, and the end rotation \psi_A = A_{HM} H_A - LP_M.
Three load cases on the top beam are supported: (1a) a concentrated transverse force W at distance a from the left joint; (1c) a concentrated moment M_o at distance a; (1d) a prescribed angular displacement \theta_o at distance a (a joint rotation, for example from a settlement or an imperfection). The output includes H_A, V_A, \psi_A, H_B, V_B and the intermediate quantities A_{HH}, A_{HM}, A_{MM}, LP_H, LP_M.
Sign convention: H_A and H_B are positive pointing left (inward); V_A and V_B are positive upward; \psi_A is positive clockwise. Load W is positive downward; moment M_o is positive clockwise; \theta_o is positive clockwise.
Model limits: linear-elastic, first-order theory — small displacements, constant E_iI_i per member, no axial deformation, no distributed load, only one load applied to the top beam at a time, and both ends pinned. The outputs are structural response quantities (reactions, end rotation); there is no pass/fail safety check.
Covered cases of this table:
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